An evidence-theory model considering dependence among parameters and its application in structural reliability analysis

Abstract Due to its strong ability to deal with the epistemic uncertainty, evidence theory has been widely used to solve complex engineering problems with very limited information. This paper proposes a novel evidence-theory model for multidimensional problems, with consideration of the dependence among evidence variables. More specifically, a joint frame of discernment (FD) is created for multidimensional problem through an ellipsoidal model , where the correlativity between the parameters is well reflected by the shape of the ellipsoid. An approach to construct reasonable ellipsoids using the experimental samples is also provided. Combination of the ellipsoidal model and the marginal basic probability assignments (BPAs) of the parameters then generates a joint BPA structure. Based on the new evidence-theory model, a reliability analysis method is formulated to evaluate the safety degree of structures with dependent epistemic uncertainty. Four numerical examples are investigated to demonstrate the applicability of the proposed method.

[1]  Jon C. Helton,et al.  Investigation of Evidence Theory for Engineering Applications , 2002 .

[2]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[3]  Leslie George Tham,et al.  Robust design of structures using convex models , 2003 .

[4]  Xiaoping Du,et al.  Reliability sensitivity analysis with random and interval variables , 2009 .

[5]  W. Dong,et al.  Vertex method for computing functions of fuzzy variables , 1987 .

[6]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[7]  Ramana V. Grandhi,et al.  Gradient projection for reliability-based design optimization using evidence theory , 2008 .

[8]  C. Jiang,et al.  A New Uncertain Optimization Method Based on Intervals and An Approximation Management Model , 2007 .

[9]  I. Elishakoff,et al.  Nonprobabilistic, convex-theoretic modeling of scatter in material properties , 1994 .

[10]  John E. Renaud,et al.  Uncertainty quantification using evidence theory in multidisciplinary design optimization , 2004, Reliab. Eng. Syst. Saf..

[11]  Scott Ferson,et al.  Problem Formulation for Probabilistic Ecological Risk Assessments , 2010 .

[12]  Xu Han,et al.  An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method , 2007 .

[13]  Su-huan Chen,et al.  Interval optimization of dynamic response for uncertain structures with natural frequency constraints , 2004 .

[14]  M. B. Anoop,et al.  Application of fuzzy sets for estimating service life of reinforced concrete structural members in corrosive environments , 2002 .

[15]  Z. Mourelatos,et al.  A Design Optimization Method Using Evidence Theory , 2006, DAC 2005.

[16]  George J. Klir,et al.  Generalized information theory: aims, results, and open problems , 2004, Reliab. Eng. Syst. Saf..

[17]  Ramana V. Grandhi,et al.  Sensitivity analysis of structural response uncertainty propagation using evidence theory , 2002 .

[18]  Ramana V. Grandhi,et al.  Epistemic uncertainty quantification techniques including evidence theory for large-scale structures , 2004 .

[19]  Xu Guo,et al.  Confidence extremal structural response analysis of truss structures under static load uncertainty via SDP relaxation , 2009 .

[20]  G. Klir,et al.  Uncertainty-based information: Elements of generalized information theory (studies in fuzziness and soft computing). , 1998 .

[21]  Z. Kang,et al.  Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model , 2009 .

[22]  Gui-Rong Liu,et al.  The optimization of the variable binder force in U-shaped forming with uncertain friction coefficient , 2007 .

[23]  Ramana V. Grandhi,et al.  Comparison of evidence theory and Bayesian theory for uncertainty modeling , 2004, Reliab. Eng. Syst. Saf..

[24]  Xiaoping Du,et al.  Uncertainty Analysis With Probability and Evidence Theories , 2006, DAC 2006.

[25]  Huixin Guo.  Method of reliability design optimization using evidence theory and interval analysis , 2008 .

[26]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[27]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[28]  Kari Sentz,et al.  Combination of Evidence in Dempster-Shafer Theory , 2002 .

[29]  I. Elishakoff,et al.  Probabilistic interval reliability of structural systems , 2008 .

[30]  Hasan Kurtaran,et al.  Crashworthiness design optimization using successive response surface approximations , 2002 .

[31]  Z. Qiu,et al.  Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis , 2005 .

[32]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[33]  W E Vesely,et al.  Fault Tree Handbook , 1987 .

[34]  F. O. Hoffman,et al.  Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. , 1994, Risk analysis : an official publication of the Society for Risk Analysis.

[35]  Zhan Kang,et al.  Structural reliability assessment based on probability and convex set mixed model , 2009 .

[36]  C. Jiang,et al.  Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique , 2011 .

[37]  Ramana V. Grandhi,et al.  An approximation approach for uncertainty quantification using evidence theory , 2004, Reliab. Eng. Syst. Saf..

[38]  Armando Mammino,et al.  Determination of parameters range in rock engineering by means of Random Set Theory , 2000, Reliab. Eng. Syst. Saf..

[39]  Jon C. Helton,et al.  Mathematical representation of uncertainty , 2001 .

[40]  Jon C. Helton,et al.  An exploration of alternative approaches to the representation of uncertainty in model predictions , 2003, Reliab. Eng. Syst. Saf..

[41]  Xiaoping Du,et al.  Sensitivity Analysis with Mixture of Epistemic and Aleatory Uncertainties , 2007 .

[42]  J. Kacprzyk,et al.  Advances in the Dempster-Shafer theory of evidence , 1994 .

[43]  Jon C. Helton,et al.  Sensitivity analysis in conjunction with evidence theory representations of epistemic uncertainty , 2006, Reliab. Eng. Syst. Saf..